First, we choose the geometric point \(\bar{s}\in S\)  such that \(\deg (\mathcal {O}_{X_{\bar{s}}}(1)):=\max \lbrace \deg (\mathcal {O}_{X_{s}}(1))|s\in S\rbrace \)  as in the proof of [[1]}, Lemma 4.26]. By Kleiman criterion for stacks in [[1]}, Theorem 4.12] for the \(S\) -flat coherent sheaf \(\mathcal {E}\)  on \(\mathcal {X}\)  as in the beginning of this subsection, one can choose an integer \(\widetilde{m}>0\)  such that \(\pi _{*}\mathcal {E}nd_{\mathcal {O}_{\mathcal {X}}}(\mathcal {E})(\widetilde{m})\)  is generated by global sections, which is certainly true on each fiber of \(X\rightarrow S\) .
For the case when \(\deg \delta <\deg P\) , one can apply the argument in the proof of [[3]}, Lemma 3.1] to get the upper bound (a function of \(s\) ) for each geometric point \(s\)  of \(S\) . With the choices of \(\bar{s}\)  and \(\widetilde{m}\) , one can
get a uniform upper bound for \(\lbrace \hat{\mu }_{\mathrm {max}}(F_{\mathcal {E}_{s}}(\mathcal {F}_{s}))| s\in S\rbrace \)  depending on the fixed \(S\) -flat sheaf \(\mathcal {F}_{0}\)  and \(P\) . When \(\mathrm {deg}\,\delta \ge \mathrm {deg}\,P\) , similarly, follow the argument in
the proof of [[3]}, Lemma 3.2] to obtain the lower bound for every geometric point \(s\)  of \(S\) , where we apply [[5]}, Lemma 2.12] to \(\hat{\mu }_{\mathrm {min}}(\mathcal {O}_{Y_{s}})\)  with \(Y_{s}\)  being the scheme-theoretic support of \(F_{\mathcal {E}_{s}}(\mathcal {F}_{s})\) . Using the flatness of \(F_{\mathcal {E}}(\mathcal {F})\)  over \(S\)  by [[1]}, Corollary 1.3-(3)] or Lemma REF , one can obtain a uniform lower bound for \(\lbrace \hat{\mu }_{\mathrm {min}}(F_{\mathcal {E}_{s}}(\mathcal {F}_{s}))| s\in S\rbrace \)  depending on \(P\)  and \(X\) . Since bounding \(\hat{\mu }_{\mathrm {min}}\)  from below is equivalent to bounding \(\hat{\mu }_{\mathrm {max}}\)  from above and the uniform bound is independent of choices of \(\mathcal {F}\)  in the set-theoretic family, the proof is completed by [[1]}, Theorem 4.27-(1)].
